fractal

I was shopping at Golden Orchard, the organic greengrocer in St. Lawrence Market where I buy my fruits and vegetables, and I saw, on the shelf, something labelled Romanesco broccoli. I couldn’t believe what I was seeing. Instead of the usual bumpy shape of broccoli, this had cones that were perfect spirals of cones that were perfect spirals of cones that were perfect spirals of cones that were… I had to have it, this fractal broccoli, this fractali, this fraccoli, this… beautiful vegetable.

I was, in a way, sad to break it apart and steam it, but that’s what it was there for: if I had left it untouched, it would have wilted and lost its beauty anyway. Steamed, it was delicious, sweet, mild, more like cauliflower in flavour – which is right enough, as it turns out actually to be more of a cauliflower, although the difference between broccoli and cauliflower is not as clear-cut as you might think.

But this is not a vegetable tasting note. I am here to talk about the word that described this vegetable’s form, a word I have already used: fractal. The odds are fairly good that you have encountered this word before, and as likely as not you have some image in your head of swirling geometric shapes that are emblematic of the beauty of mathematics in some ineffable way. They have perhaps a fragile beauty in their tracery, or perhaps a spectacular one; you may see them as evidence of craft or as something quite the opposite; or you may see them as infractions of your sense of order, causing you to rack your brain to the point of fracture. Your taste of this word will follow.

But the concept of the fractal is something other than what most people think it is, and at root involves an elegant simplicity. Ah, yes, another term mathematicians love – elegant. Ordinary people think of silver and china and linen and formal wear; mathematicians think of power in simplicity.

Fractal comes from the Latin fractus, broken. This may seem odd for things that appear to be very complex but entirely connected. But what they are is breakable: break one into smaller parts and you will have smaller parts that look like a whole breakable into smaller parts. It is self-similar recursivity: it contains a replica of itself, which contains a replica of itself, and so on. It’s the hall of mirrors.

I’m reminded of the Apple Lisa computer – the computer (a precursor to the Macintosh that did not meet with great success) was actually named after Steve Jobs’s illegitimate daughter, but the marketing explanation was that it stood for local integrated systems architecture. Now, you might think that that is somehow a definition for recursivity, but what I’m coming to is that the in-joke among the engineers working on the computer was that it stood for Lisa: invented stupid acronym. The first word of Lisa: invented stupid acronym is of course Lisa, which stands for Lisa: invented stupid acronym, and so on.

Some other examples of self-similar recursivity are in order. An easy one is Russian dolls: inside each one is a smaller one, and so on; in the world of math, unhampered by the limits of materials, these could continue all the way down to the infinitely small, and even at the infinitely small be just like the original size, with an infinity of nested smaller parts. This is what those swirling geometric shapes do: at whatever magnification, you see something much like the original, a shape having smaller echoes of the shape that have smaller echoes of the shape and so on. You see this in the Romanesco broccoli.

Adding the dimension of time, you can even see it in humans. A girl is born, grows, becomes a woman; then she grows a child within her, which is born, grows, becomes an adult; and so on. And in the greater web of life, what we eat feeds the future, as it has been fed by the past. Is each generation less than the previous? No – with infinite recursive self-similarity, the idea of absolute scale is meaningless. To quote Walt Whitman’s Song of Myself,

Before I was born out of my mother generations guided me,
My embryo has never been torpid, nothing could overlay it.

For it the nebula cohered to an orb,
The long slow strata piled to rest it on,
Vast vegetables gave it sustenance,
Monstrous sauroids transported it in their mouths and deposited it
with care.

All forces have been steadily employ’d to complete and delight me,
Now on this spot I stand with my robust soul.

Something else that bumps up against the limits of meaning is how many dimensions a fractal has. Consider one of those fractal illustrations you see. It’s made of a line with curves and angles. If you zoom in on the curves and angles, each one has the same curves and angles within it, but smaller. And so on. Now, this shape is just a line, so it has no area; in math, a line is simply one-dimensional with no thickness, only length. But say you try to measure its length. You might look and see what appears to be a measurable bit of line with curves. But inside each curve is a smaller curve that adds some more length, and inside each smaller curve is a smaller one that adds a little more length, and so on to the infinity of smallness. While material limitations prevent something like a vegetable from actually continuing to infinite smallness, an abstract line of this sort has no such limitation, and within each curve is an infinity of smaller curves, meaning that the line has infinite length, and every segment of it has infinite length. In one dimension it is too full; in two, it is too empty.

If the idea that a fractal could have zero area but infinite length doesn’t make sense to you, then good. It shouldn’t. It means we’re not measuring it the right way. Consider: if you ask about the area of a line or the volume of a square, you get zero, because you’re measuring it using too many dimensions. But if you try to get the measure the volume of a cube using two-dimensional squares, or the area of a square using one-dimensional lines, you will get infinity, because a square has no thickness in the third dimension, and a line has no thickness in the second. So one dimension is not enough dimensions in which to measure the fractal line, but two is too many. It actually exists with fractional dimensionality: 1.25, 1.33, 1.5… depending on the specific pattern.

It also stands to reason that my beautiful Romanesco broccoli, which is a three-dimensional object, represents (imperfectly, because of physical limitations) a shape that has more than three dimensions, and less than four.

I will stop there before I fracture the heads of any of my readers, but if you are intrigued by this, I recommend Yale University’s site on fractal geometry, which gives very clear and friendly explanations without dumping too much on you at once. A warning, though: to pursue a topic, you will find that you click a link, and that page has subtopics so you click on a link, and that page has subtopics so you click on a link…

Which brings me to the point that the World Wide Web, with its hyperlinks in hyperlinks in hyperlinks, is also, in its way, fractal. That’s right. You are reading this on a fractal that exists (by the calculation of J.S. Kim et al., arXiv:cond-mat/0605324v1) in 4.1 dimensions.

And you, too, of course, are part of a fractal. Like Walt Whitman, you are large, you contain multitudes, just as you and Walt are contained within the same fractal web in time, in the fractal folds of time. And you say, with Whitman,

The past and present wilt – I have fill’d them, emptied them.
And proceed to fill my next fold of the future.

The original fractal image generator, Fractint, is still going 23 years after the first version. I had one in the dear dead days of DOS. It now comes in Windows, Mac and Linux flavours. These two pages lead to the downloads, in a rather convoluted and nerdish way; the second has the Mac version.http://www.fractint.org/http://www.nahee.com/spanky/www/fractint/

I’m sorry, I have to disagree with this:
> meaning that the line has infinite length

Actually, each further segment down the spiral also is smaller, until it’s infinitely small. I’m sure this instantly reminds you of the concept of the “limit”. :)
I don’t know the exact formula for a fractal spiral, but there definitely is a “lim (x→0) (…)” in there. So the sum is not infinite.

I see I am open to accusations of Zeno-phobia. :P I prefer not to speak on my own authority with regard to the question of infinite length; I turn to http://classes.yale.edu/fractals/ for backup, where, you will see, it readily declares that a Koch curve, for instance, is “infinitely long.”

Off the top of my head, a thing we need to remember is that normally when we speak of limits we are talking about things such as straightforward curves, which we are just subdividing smaller and smaller. As you divide a curve of the more usual sort, or a line or other function, into smaller and smaller segments, each division into half truly does result in a length half as long; we can see readily that the distance Zeno’s rabbit travels is measurable macroscopically with accuracy. But with a fractal such as a Koch curve or the Mandelbrot set, at each level of magnification you discern still more self-similar curves; you can continue to an infinity of magnifications and there will still be an infinity of magnifications available with yet more curve detail. It is true that these curves will be smaller and smaller and smaller, but the distinction is that at every refinement of resolution you add more length. A line seen at a 200% magnification displays 1/2 the length. A fractal curve seen at 200% magnification will display more than 1/2 the length, without limit as you move through greater magnifications to infinity.

Say you have a curve that at first appears to have a length of 1, but on closer inspection you see that it has details that make the length 1.2. Then you magnify it and you discover that any given half of it has length of not 0.6 but 0.72. And Then you magnify again and discover that each half of a half has a length of not 0.36 but 0.432. Simply by magnifying to 400% you have increased your estimation of the line’s length from 1 to 1.728. You aren’t adding 1/2+1/2=1/4+1/4+1/4+1/4=… ; you find out you’re actually discovering more length with each refinement in resolution. The problem is actually more like figuring compound interest.

In real life, of course, there are limits to resolution; the curve stops being self-similar at a certain level of magnification. Only in our brains do lines have zero thickness. But in principle, fractals are infinite in one dimension and zero in the next, which is why they’re described as having fractional dimensionality. And even in real life, the effect shows up to a bedeviling extent – the famous problem of the length of a shoreline: with how much resolution do you measure it? The more resolution, the longer it works out to be. A smooth curve approaches a limit, as you say, but a fractal curve is limitless in principle.

But of course if there’s something I’m overlooking here, I’m sure someone will point it out. I’m not a mathematician by profession.

On the Yale site, they have it set up as a frameset, so the URL just takes you to the main page. Go to “Ineffective ways to measure.” Follow its approach to measuring a Koch curve. In the case of the Koch curve, each successive refinement of measurement multiplies the previous estimate by 4/3, as the middle segment of each line is in fact replaced by two segments that would form an equilateral triangle with the segment they replace as base. So at each refinement of measurement the length multiplies by 4/3.

With n refinements of measurement, the length of a segment is 1/3^n, but the number of segments is 4^n (not 3^n as it would be with a line of dimension=1). After 128 refinements of measurement, a curve with a horizontal extent of 1 metre will have a segment length of 8.5(10^-62) m and a total measured length of approximately 1 light year.

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